Detection of continuous-time quaternion signals in additive noise

Jesús Navarro-Moreno,José M Quesada-Rubio,Juan Carlos Ruiz-Molina,Antonia Oya

doi:10.1186/1687-6180-2012-234

Jesús Navarro-Moreno, José M Quesada-Rubio + Show 2 more

Open Access

https://doi.org/10.1186/1687-6180-2012-234

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Abstract

Different kinds of quaternion signal detection problems in continuous-time by using a widely linear processing are dealt with. The suggested solutions are based on an extension of the Karhunen-Loeve expansion to the quaternion domain which provides uncorrelated scalar real-valued random coefficients. This expansion presents the notable advantage of transforming the original four-dimensional eigen problem to a one-dimensional problem. Firstly, we address the problem of detecting a quaternion deterministic signal in quaternion Gaussian noise and a version of Pitcher’s Theorem is given. Also the particular case of a general quaternion Wiener noise is studied and an extension of the Cameron-Martin formula is presented. Finally, the problem of detecting a quaternion random signal in quaternion white Gaussian noise is tackled. In such a case, it is shown that the detector depends on the quaternion widely linear estimator of the signal.

Highlights

Quaternion signals are of great relevance to applications in the area of statistical signal processing in which the received signal is composed of a certain number of random components since they account naturally for their correlated nature [1,2]
We demonstrate that the log-likelihood ratio depends on the quaternion widely linear (QWL) estimator of the signal provided in [12]
We study the detection problem of a quaternion random signal in additive quaternion white Gaussian noise (QWGN), i.e, we consider the hypotheses pair t

Summary

Introduction

Quaternion signals are of great relevance to applications in the area of statistical signal processing in which the received signal is composed of a certain number of random components since they account naturally for their correlated nature [1,2] These are useful, for example, in studying communication, electromagnetics, seismology, acoustics, etc., problems frequently encountered in this area [3]. The technique to derive the QKL expansion is based on the definition of a real-valued univariate stochastic signal whose second-order statistics match that of quaternion This strategy avoids addressing a fourdimensional vectorial problem which notably simplifies the obtaining of the representation. We address the detection of quaternion random signals in quaternion white Gaussian noise (QWGN) In this case, we demonstrate that the log-likelihood ratio depends on the QWL estimator of the signal provided in [12].

Preliminaries

Detection of quaternion deterministic signals in quaternion Gaussian noise

Particular case: the general quaternion Wiener process

Detection of quaternion random signals in QWGN

Conclusions